Egads. This is the schematic diagram we've always been shown. Does it give you the feeling of sine? Not any more than a skeleton portrays the agility of a cat. Let's watch sine move and then chart its course.

The Unavoidable Circle

Circles have sine. Yes. But seeing the sine inside a circle is like getting the eggs back out of the omelette. It's all mixed together!

Let's take it slow. In the simulation, set Hubert to vertical:none and horizontal: sine*. See him wiggle sideways? That's the motion of sine. There's a small tweak: normally sine starts the cycle at the neutral midpoint and races to the max. This time, we start at the max and fall towards the midpoint. Sine that "starts at the max" is called cosine, and it's just a version of sine (like a horizontal line is a version of a vertical line).

Ok. Time for both sine waves: put vertical as "sine" and horizontal as "sine*". And... we have a circle!

A horizontal and vertical "spring" combine to give circular motion. Most textbooks draw the circle and try to extract the sine, but I prefer to build up: start with pure horizontal or vertical motion and add in the other.

Quick Q & A

A few insights I missed when first learning sine:

Sine really is 1-dimensional

Sine wiggles in one dimension. Really. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave.

For the geeks: Press "show stats" in the simulation. You'll see the percent complete of the total cycle, mini-cycle (0 to 1.0), and the value attained so far. Stop, step through, and switch between linear and sine motion to see the values.

Quick quiz: What's further along, 10% of a linear cycle, or 10% of a sine cycle? Sine. Remember, it barrels out of the gate at max speed. By the time sine hits 50% of the cycle, it's moving at the average speed of linear cycle, and beyond that, it goes slower (until it reaches the max and turns around).

So x is the 'amount of your cycle'. What's the cycle?

It depends on the context.

Basic trig: 'x' is degrees, and a full cycle is 360 degrees

Advanced trig: 'x' is radians (they are more natural!), and a full cycle is going around the unit circle (2*pi radians)

Play with values of x here:

But again, cycles depend on circles! Can we escape their tyranny?

Pi without Pictures

Imagine a sightless alien who only notices shades of light and dark. Could you describe pi to it? It's hard to flicker the idea of a circle's circumference, right?

Let's step back a bit. Sine is a repeating pattern, which means it must... repeat! It goes from 0, to 1, to 0, to -1, to 0, and so on.

Let's define pi as the time sine takes from 0 to 1 and back to 0. Whoa! Now we're using pi without a circle too! Pi is a concept that just happens to show up in circles:

Sine is a gentle back and forth rocking

Pi is the time from neutral to max and back to neutral

n * Pi (0 * Pi, 1 * pi, 2 * pi, and so on) are the times you are at neutral

2 * Pi, 4 * pi, 6 * pi, etc. are full cycles

Aha! That is why pi appears in so many formulas! Pi doesn't "belong" to circles any more than 0 and 1 do -- pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!

Question: If pi is half of a natural cycle, why isn't it a clean, simple number?

Let's answer a question with a question. Why does a 1x1 have a diagonal of length sqrt(2) = 1.414... (an irrational number)?

It's philosophically inconvenient when nature doesn't line up with our number system. I don't have a good intuition. My hunch is simple rules (1x1 square + Pythagorean Theorem) can still lead to complex outcomes.

How fast is sine?

I've been tricky. Previously, I said "imagine it takes sine 10 seconds from 0 to max". And now it's pi seconds from 0 to max back to 0? What gives?

sin(x) is the default, off-the-shelf sine wave, that indeed takes pi units of time from 0 to max to 0 (or 2*pi for a complete cycle)

sin(2x) is a wave that moves twice as fast

sin(x/2) is a wave that moves twice as slow

So, we use sin(n*x) to get a sine wave cycling as fast as we need. Often, the phrase "sine wave" is referencing the general shape and not a specific speed.

Part 2: Understanding the definitions of sine

That's a brainful -- take a break if you need it. Hopefully, sine is emerging as its own pattern. Now let's develop our intuition by seeing how common definitions of sine connect.

Definition 1: The height of a triangle / circle!

Sine was first found in triangles. You may remember "SOH CAH TOA" as a mnemonic

SOH: Sine is Opposite / Hypotenuse

CAH: Cosine is Adjacent / Hypotenuse

TOA: Tangent is Opposite / Adjacent

For a right triangle with angle x, sin(x) is the length of the opposite side divided by the hypotenuse. If we make the hypotenuse 1, we can simplify to:

Sine = Opposite

Cosine = Adjacent

And with more cleverness, we can draw our triangles with hypotenuse 1 in a circle with radius 1:

Voila! A circle containing all possible right triangles (since they can be scaled up using similarity). For example:

sin(45) = .707

Lay down a 10-foot pole and raise it 45 degrees. It is 10 * sin(45) = 7.07 feet off the ground

An 8-foot pole would be 8 * sin(45) = 5.65 feet

These direct manipulations are great for construction (the pyramids won't calculate themselves). Unfortunately, after thousands of years we start thinking the meaning of sine is the height of a triangle. No no, it's a shape that shows up in circles (and triangles).

Realistically, for many problems we go into "geometry mode" and start thinking "sine = height" to speed through things. That's fine -- just don't get stuck there.

Definition 2: The infinite series

I've avoided the elephant in the room: how in blazes do we actually calculate sine!? Is my calculator drawing a circle and measuring it?

Glad to rile you up. Here's the circle-less secret of sine:

Sine is acceleration opposite to your current position

Using our bank account metaphor: Imagine a perverse boss who gives you a raise the exact opposite of your current bank account! If you have $50 in the bank, then your raise next week is -$50. Of course, your income might be $75/week, so you'll still be earning some money ($75 - $50 for that week), but eventually your balance will decrease as the "raises" overpower your income.

But never fear! Once your account hits negative (say you're at -$50), then your boss gives a legit $50/week raise. Again, your income might be negative, but eventually the raises will overpower it.

This constant pull towards the center keeps the cycle going: when you rise up, the "pull" conspires to pull you in again. It also explains why neutral is the max speed for sine: If you are at the max, you begin falling and accumulating more and more "negative raises" as you plummet. As you pass through then neutral point you are feeling all the negative raises possible (once you cross, you'll start getting positive raises and slowing down).

By the way: since sine is acceleration opposite to your current position, and a circle is made up of a horizontal and vertical sine... you got it! Circular motion can be described as "a constant pull opposite your current position, towards your horizontal and vertical center".

Geeking Out With Calculus

Let's describe sine with calculus. Like e, we can break sine into smaller effects:

Start at 0 and grow at unit speed

At every instant, get pulled back by negative acceleration

How should we think about this? See how each effect above changes our distance from center:

At any moment, we feel a restoring force of -x. We integrate twice to turn negative acceleration into distance:

Seeing how acceleration impacts distance is like seeing how a raise hits your bank account. The "raise" must change your income, and your income changes your bank account (two integrals "up the chain").

So, after "x" seconds we might guess that sine is "x" (initial impulse) minus x^3/3! (effect of the acceleration):

Something's wrong -- sine doesn't nosedive! With e, we saw that "interest earns interest" and sine is similar. The "restoring force" changes our distance by -x^3/3!, which creates another restoring force to consider. Consider a spring: the pull that yanks you down goes too far, which shoots you downward and creates another pull to bring you up (which again goes too far). Springs are crazy!

I saw this formula a lot, but it only clicked when I saw sine as a combination of an initial impulse and restoring forces. The initial push (y = x, going positive) is eventually overcome by a restoring force (which pulls us negative), which is overpowered by its own restoring force (which pulls us positive), and so on.

A few fun notes:

Consider the "restoring force" like "positive or negative interest". This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to learn here :).

For small amounts, "y = x" is a good guess for sine. We just take the initial impulse and ignore any restoring forces.

The Calculus of Cosine

Cosine is just a shifted sine, and is fun (yes!) now that we understand sine:

Sine: Start at 0, initial impulse of y = x (100%)

Cosine: Start at 1, no initial impulse

So cosine just starts off... sitting there at 1. We let the restoring force do the work:

Again, we integrate -1 twice to get -x^2/2!. But this kicks off another restoring force, which kicks off another, and before you know it:

Definition 3: The differential equation

We've described sine's behavior with specific equations. A more succinct way (equation):

This beauty says:

Our current position is y

Our acceleration (2nd derivative, or y'') is the opposite of our current position (-y)

Both sine and cosine make this true. I first hated this definition; it's so divorced from a visualization. I didn't realize it described the essence of sine, "acceleration opposite your position".

And remember how sine and e are connected? Well, e^x can be be described by (equation):

The same equation with a positive sign ("acceleration equal to your position")! When sine is "the height of a circle" it's really hard to make the connection to e.

One of my great mathematical regrets is not learning differential equations. But I want to, and I suspect having an intuition for sine and e will be crucial.

Summing it up

The goal is to move sine from some mathematical trivia ("part of a circle") to its own shape:

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146 Comments on "Intuitive Understanding of Sine Waves"

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uwe

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This was excellent! Well done.

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6 years 3 months ago

Kalid

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Another great article from master Kalid, I’m really happy :D.

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6 years 3 months ago

Erich

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Excellent work! Thank you.

I particularly enjoyed having the infinite series model click intuitively, and seeing that the unit circle contained all possible right triangles. Why, yes, yes it does!

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6 years 3 months ago

Anonymous

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Brilliant. I must agree with Erich, the infinite series visualization is wonderfully intuitive.

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@D-POWER: Awesome ;)

@Erich/Anonymous: Thanks for letting me know what made it click! I’m working on an idea to make it easier to share these types of aha moments.

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6 years 3 months ago

Polyergic

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I guess this would be the numbers aspect of what i saw won’t with it lol

I mentioned this in my comment, please check it out! Lemme know if this relates to what i was talking about. I’m really not good with numbers, so.

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@Polyergic: Doh! Great catch — it should be fixed now :).

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6 years 3 months ago

Anonymous

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Wow! Awesome Mr Kalid . You really should have been my teacher :)

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6 years 3 months ago

Anonymous

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You have no idea how happy you just made me.

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6 years 3 months ago

Anonymous

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@Anon1: Thank you!

@Anon2: Glad it helped — sine has bugged me for so long.

@Anon3: Love those pictures! Our brains need both :).

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6 years 3 months ago

Anonymous

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@Anonymous: That’s just the names we’ve given to those ratios, like saying perimeter = 4 * side [in a square].

But as it turns out, sine isn’t limited to triangles — that is just the first place it was noticed.

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6 years 3 months ago

Anonymous

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I haven’t made the connection between sine as an idea and why the ratios in SOHCAHTOA are what they are. Am I making sense?

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6 years 3 months ago

nschoe

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Wow, thanks once again Kalid. Your explanatiosn are truly wonderful, just how do you come to such a level of knowledge and how do you manage to explain it so easily?
I wish you were my teacher.
Every article is just magic, please keep writing it’s a real relief every time you release another article.

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6 years 3 months ago

loimprevisto

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Thank you!

That was a fantastic lesson. Since I left school I’ve come back to math every few years to try and remember everything I’d forgotten. The best feeling in the world (yep, even better than *that* one…) is the “Eureka!” moment when everything just makes sense. Your article gave me two of those, from watching Hubert move in his circle and from seeing the derivitave definition of sine and how it related to e. You have a gift for teaching and writing, thank you for sharing it.

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6 years 3 months ago

Kalid

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@nschoe: Thanks for dropping by! I appreciate the kind words — I don’t think I really understand that much, it’s more my lack of understanding/satisfaction which drives me to seek simpler explanations. The notion that sine is this cyclical wave that we all see just didn’t click deeply with me, I needed something deeper. Many ideas are like that (e, imaginaries, etc.) so I start trying to find analogies that might fit better :).

I’ll definitely keep writing, appreciate the support!

@loimprevisto: You’re welcome! You got it, those Eureka moments are so incredibly fulfilling. It’s what I strive for when writing, I just want to share what clicked hoping it clicks for other people too. Thanks for sharing what aspects helped (Hubert / derivative definition), I have a project in the works to make these insight exchanges easier & more community driven :).

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[…] Intuitive Understanding of Sine WavesI’m generally a fan of Better Explained, but this is an especially good article. […]

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6 years 3 months ago

Anon

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Since my engineering studies I always liked Euler’s formula, connecting sine and cosine to the unit circle in the complex number plane. That’s what Hubert’s sine-sine setting reminded me of.

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6 years 3 months ago

Kalid

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I’ve been reading these for a while, I have to say I think this is the best one yet. We did Taylor series a month or so ago in my Calc class, the end of this article aided my comprehension a whole lot more than any of the class work ever did. Keep ’em coming, please!

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@Joe: Thanks — this was one of the longer ones to write so glad it was helpful… I’ll keep cranking :)

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6 years 3 months ago

mark ptak

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You are a master. At some point I would love to hear your take on why the pattern that emerges in transformation matrices
cos -sin
sin cos
changes for rotations about the y axis. For now I”m feeling hungry for a salad….

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